Dear Family, your Operation Maths guide to Lines and Angles

Digging Deeper into … 2-D Shapes and 3-D Objects (infants to second class)

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes and 3-D objects, please check out the following posts:

Dear Family, your Operation Maths Guide to 2-D Shapes

Dear Family, your Operation Maths Guide to 3-D Objects

3-D shapes or 3-D objects?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe both 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is titled 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the 1999 Primary Maths curriculum.

So what first? 2-D or 3-D?

2-D and 3-D objects are very inter-related, to the point that there is often much debate about which of the topic should be taught first; 2-D shapes, 3-D objects or teach them both concurrently.

Since 2-D shapes are lacking the third dimension of depth or height that their 3-D relations possess, this makes them quite abstract as only flat, drawn/printed shapes are truly 2-D. Whereas, 3-D objects can be picked up, manipulated, used for constructions etc., making them much more suited to the concrete learning experiences that are essential in the early years. They are the objects that we find in the real-world. Thus, since 2-D shapes are only flat representations of the faces of 3-D objects, it could be argued that it would be more logical, and more in line with the concrete-pictorial-abstract (CPA) approach, to teach about 3-D objects before 2-D shapes.

On the other-hand, it could be argued that 2-D shapes should be taught first as it is likely than young children would be more familiar with them. For example, the vocabulary of 2-D shapes features more regularly in common speak than the vocabulary of 3-D objects. Many children will likely have encountered many 2-D shapes from picture books and patterns around their homes, etc. And so, it remains inconclusive as to which order of progression is most beneficial!

In the Operation Maths books, the children meet the specific topic of 2-D shapes prior to that of 3-D objects each school year. However, it is envisaged that by the time the children in the junior classes are formally engaging with 2-D shapes, they have already encountered and informally explored both 2-D shapes and 3-D objects via the monthly themes (laid out in the long-term and short-term plans of each TRB) and in the suggested Aistear play activities (detailed also in the TRB) of which, the Aistear theme of construction is particularly relevant.

Infant classes

Whether considering 2-D shapes or 3-D objects, the suggested progression within each topic is very similar:

• Undirected play
• Sorting and ordering activities
• Building and making (including making patterns)
• Identifying

Undirected play may include sand and water play, use of formal construction toys, constructing using “junk” or found materials; any activities that allow the children to handle and manipulate shapes and objects.
In the Operation Maths TRBs for junior and senior infants there are ample suggestions for suitable activities, under the headings of various themes. “Undirected play” does not imply that the teacher is superfluous to the process; rather while the children are the instigators, the teacher can play a vital role, observing the way in which the children interact with the materials, and asking the children to explain what they did, how they did it and why they did it that way. This can be a great way to assess the prior knowledge and language that the children may already have.

Sorting and ordering activities include the Early Mathematical Activities (EMA) used early on in the infant classes; thus it is likely that shapes and/or objects have already been used as part of these activities, for example sorting and matching according to colour, size etc; ordering according to length/height etc.

At this point, the children should also be prompted to sort the shapes and objects according to their respective properties as relevant and appropriate:

• Sort 2-D shapes according to the number of corners and the number and type of sides (straight, curved or both; sides that are different or the same).
• Sort 3-D objects according to those that roll/do not roll, slide/do not slide, build/do not build, are hollow/solid; as a development, according to the number of corners and the number and type of faces and edges (please see end of post for more information on faces and edges).
• The teacher can also isolate shapes/objects to create sets and then ask the children to identify the rule of the set: “What’s my rule?” (see image above). The children can also be encouraged to play the “What’s my rule?” game in groups.
• Isolate a particular shape/object in the room and ask the children to locate others that are the same/similar and make a set of like objects/shapes.
• The children may also be naming the shapes as part of these explorations; however this is not necessary as it is more important that they appreciate the similarities and difference between shapes, rather than identifying them.

Building and making with shapes/objects may have already been explored informally as part of the undirected play phase. The purpose now is to develop this into more formal teacher-directed tasks and activities:

• Build the tallest building/castle that you can. What objects did you use/not use and why?
• Dip a face of a 3-D object in paint and use it to print. Make a pattern using the prints. What do you notice?
• Try printing with different faces of the same 3-D object. Are the resulting prints the same or different?
• Push 3-D objects into sand/plasticine to make imprints. Or (if able) trace around the 3-D objects on paper to make designs.
• Use cut-out shapes, gummed shapes, tangrams and/or pattern blocks to make pictures and patterns.
• Use the shapes to cover surface of your book/mini-white board; which shapes did you use/not use and why?
• Combine two shapes/objects to make a new shape/object.

As part of the building and making activities the children may begin to realise how certain shapes/objects can be combined to become other shapes/objects. Similarly, through the reverse of these activities, and other shape cutting activities, the children should begin to realise that shapes can also be separated (partitioned) to reveal new shapes. This can include deconstructing 3-D objects to reveal their net. These activities can be revisited once the children can also name the shapes/objects, so as to arrive at certain understandings and become more accurate with mathematical language eg that two squares can make a rectangle; that, when using tangrams, two of the same size triangles can be rearranged to make a square, a larger triangle etc.

Identifying the specific shapes/objects evolves from the previous activities as the children begin to realise that it is the specific properties and attributes of a shape/object that defines it, eg all shapes with three corners (and three sides) are triangles, irrespective of their size or colour and irrespective of the measure of sides and corners (later to be referred to as angles). Activities which will serve to reinforce this include “Guess the shape/object” using descriptions (see below), guessing unseen shapes/objects from touch (eg in a feely bag), locating a specific shape from a collection using touch alone.
Through the experiences of printing and imprinting with the 3-D objects, it is also hoped that the children realise that the flat faces of 3-D objects are in fact 2-D shapes.

First and Second classes

The children in these classes will continue to sort, describe, compare and name shapes as done in infants, but to now also include new shapes and objects i.e. semi-circle (1st), oval and cone (2nd). They will continue to construct and make shapes, extending this to creating and drawing the shapes themselves.

They will further explore the combining and partitioning of 2-D shapes, and this understanding will extend to include the fractions of halves (1st) and quarters (2nd). The properties of 2-D shapes will be further explored
in second class via the stand units of symmetry, angles and area (i.e. tessellating 2-D shapes)

Further Reading and Resources:

• Dear Family, your Operation Maths Guide to 2-D Shapes and Dear Family, your Operation Maths Guide to 3-D Objects include practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand unit. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.
• Shape and Space Manual from PDST
• 3-D objects Pinterest board
• 2-D shapes Pinterest board
• Digging Deeper into 3-D objects (third to sixth class)
• Digging Deeper into 2-D shapes (third to sixth class)

Digging Deeper into … 3-D Objects

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 3-D objects, please check out the following post: Dear Family, your Operation Maths Guide to 3-D Objects

The first obvious question to be addressed is why this topic is called 3-D objects in Operation Maths, when most other textbooks, and even the curriculum refer to this topic as 3-D shapes?

In the PDST Shape and Space manual, it is suggested that “using the word ‘shape’ to describe 2-D shapes and 3-D shapes can cause confusion for pupils”. For example, asking pupils to ‘describe the shape of this shape’ highlights one problem. Another problem is that pupils must be able to think of all cuboids as being ‘the same shape’, while mathematically speaking all cuboids are not the same shape.

The manual goes on to suggest that it would be more helpful to refer to 3-D things as ‘objects’. Using ‘objects’ also reinforces the notion that if it can be physically handled/picked up, it must be a 3-D object, as opposed to a 2-D shape which should always only have length and width, not depth/height.

So, throughout the Operation Maths books, this topic is called 3-D objects to avoid confusion and to provide clarity for the pupils. However, wherever there is reference to “strand unit”, the term 3-D shapes is used, as this is the term used in the curriculum.

CPA

As usual, this topic is explored using a CPA approach, where the initial focus is on the exploration of 3-D objects and the identification of similarly shapes objects from the child’s environments. And, in a similar way to the teaching of 2D shapes, while the 3-D objects will be identified by name, the greater focus should be on their properties, as appropriate to each class level e.g.

• identifying whether the objects roll, stack, or slide
• relating the properties of each object to its purpose and use in the environment
• recording, sorting and comparing according to the number and shape of faces and curved surfaces
• recording, sorting and comparing according to the number of edges and vertices (corners)

It is important that the children discover the properties of the 3-D objects by hands-on investigations and by classifying and sorting. Sorting activities help develop the children’s communication, observation, reasoning and categorising skills, and thus will help to develop a conceptual understanding of the objects. In the senior classes using online interactive sorting activities based on Venn diagrams or Carroll diagrams can be very useful. Asking children to bring in examples of 3-D objects from home will help them to become aware of 3-D objects in their environment.

Faces, curved surfaces and edges

How many faces on a cylinder? Three or two?

Traditionally, in Ireland, and in Irish textbooks, a cylinder was recorded as having three faces. However, this is not mathematically correct, as strictly speaking a face is flat, and a 2D shape (figure), so therefore a cylinder has in fact only two faces, (both circles), and one curved surface. And while some may argue that a cylinder has a third face i.e. the rectangular shape you see when you disassemble the net of the 3-D object, in this disassembled state it is no longer a cylinder, since it can no longer roll, a specific property of all cylinders.

Another way to think about the faces of 3-D objects is to consider the number and shape of the resulting outlines of tracing around each surface of the 3-D object. It is only possible to trace around the opposite ends/bases of the cylinder, since only these are flat, thus it has only two faces, both of which are circular in shape. Similarly, it is only possible to trace around one surface on a cone, which therefore means it has only one face (a circle) and one curved surface.

And how many edges on a cylinder? Officially none, as an edge is where two flat faces meet and the faces on a cylinder are on opposite sides and do not touch/meet. However, that leaves the problem of how to describe the place where each face meets the curved surface.  So in Operation Maths, as occurs typically in other primary texts in other countries, there is a distinction made between straight edges (which are in fact true edges) and curved edges (which strictly speaking are not edges).

In 5th and 6th class, it is important that the children realise that only some of the 3D objects are also polyhedra, and that only five of these are categorised as platonic solids.

Further Reading and Resources

• Dear Family, your Operation Maths Guide to 3-D Objects includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• Virtual Maths Manipulatives for Shape and Space: Lots of tools that can be used in many different ways to explore 3-D objects and other shape concepts.
• Shape and Space Manual from PDST
• NRICH: selection of problems, articles and games for 3-D Objects
• 3-D objects Pinterest board

Digging deeper into … the Circle (5th & 6th)

While the circle, as a topic, is not a specific strand unit in itself, it traditionally has been dealt with separately, and more in depth, in 5th and 6th class. It is in these classes that the children begin to explore the circle as something more than just a 2D shape, and rather as a shape which has specific parts that can be named (eg diameter, radius, etc) and specific properties that can be explored. Fifth class is also the first time that the children have encountered degrees as a measurement of rotation (they first meet degrees in the strand unit of lines and angles). Fifth class is also the first occasion when the children would be required to use degrees to construct pie-charts in the strand unit of representing and interpreting data, for which the children require knowledge of constructing circles and dividing them into sectors prior to creating pie charts.

Thus, in Operation Maths, this chapter is placed after the chapters of Lines and Angles and 2-D Shapes and before the chapter of Data.

CPA Approach

As is typical in Operation Maths, A CPA approach is taken to this topic where the emphasis at the introductory stages is on the children exploring and examining circles in their environment and then using the manipulation of cut-out circles to identify the parts of circle and label them using the correct mathematical terminology. Indeed the children themselves can be used to model a circle, as outlined below. This type of activity will also greatly suit the kinaesthetic learners.

When looking for circles in the environment, it is also important to accurately identify circles, which are examples of 2D shapes, as distinct from cylinders, which are examples of 3D objects. So, if a child suggests that a coin is an example of a circle, emphasise that the face of a coin is indeed a circle, but that the coin itself is a cylinder.

Integrating maths and literacy

When introducing this new terminology, ask the children to suggest examples of words with similar prefixes/rootwords so as to foster connections and deepen meaning eg diameter coming from dia meaning across, through and thus connected with diagonal, diaphragm, dialogue; radius as being related to radiate, radio, radar (originating from a central point and moving outwards), etc. Etymology websites, such as Etymonline can be very useful to research and collect related words.

Measuring and constructing circles

The children should be provided with ample opportunities to measure the radius and diameter of circles of various sizes and, in doing so, be guided to discover for themselves that the measure of the diameter is twice the radius.

In a similar way, in 6th class, through comparing the measurement of the circumference and diameter of various circles, it is hoped that the children realise that the circumference of a circle is always just over three times the measure of the diameter. They can explore this in a very concrete way by measuring the circumference of various circles using string/wool and then cutting the string into lengths that equal the diameter, as show in the image below from K-5 Math Teaching Resources

From K-5 Math Teaching Resources

Again before constructing circles using a compass, the children should be asked to suggest ways to draw circles that they may have used previously and to identify the pros and cons of these methods. When ready to use a compass, it can be a good idea to use a video to demonstrate, such as the ones below.

Circles in art

Once the children have mastered the basics of constructing a circle, they could be encouraged to look and respond to circles in art, for example the work of Kandinsky, Anwar Jalal Shemza or the multitude of circle themed pieces available to view on the internet (just search google images for circles in art). The children could even look and respond to crop circles (in particular the activities of John Lundberg) or the use of circles in famous brand logos. For more ideas, check out this Pinterest board which includes circle themed art lessons. Using circles in such a way provides purposeful opportunities to use and reinforce the specific circle terminology e.g. diameter, circumference, arc, etc.

Further Reading:

Shape and Space Manual from PDST, p.154-157

Digging Deeper into … 2D Shapes (3rd – 6th)

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of 2-D shapes, please check out the following post: Dear Family, your Operation Maths Guide to 2-D Shapes

Overview of 2D-Shapes:

The following are the new 2D shapes, to which the children are formally introduced, in the senior end of primary school:

• 3rd class: hexagon
• 4th class:  Equilateral, isosceles and scalene triangles; rhombus & parallelogram; pentagon and octagon
• 5th class: Polygons, quadrilaterals, trapezium
• 6th class: Kite*

* while the kite is not specified on the Primary Mathematics Curriculum (1999) for sixth class, it has been included in Operation Maths 6 as it features on the curricula for 5th/6th grade in many other countries.

As with every topic in Operation Maths, a CPA approach is also recommended for 2D shapes:

Concrete: Using concrete geometric shapes for classifying and sorting; identifying examples of 2D shapes and tessellations in the environment.
Pictorial: Tracing around shape templates to make reproductions that can be manipulated, folded, partitioned and combined; using lollipop sticks, geostrips or geoboards to create representations of 2D shapes; using the Operation Maths Scratch lessons accessible on edcolearning.ie to draw various shapes.
Abstract: Answering shape questions with no visual references/supports; suggesting the number of lines of symmetry on a shape without folding or drawing to explore the same; identifying the resulting shape when a given shape is rotated, flipped etc.

Properties of 2D Shapes:

Don’t let the list above, of 2D shapes by class, fool you; it shouldn’t create an incorrect impression that the primary focus is on identifying shapes, or that we should look at one type of 2D shape exclusive of others. Rather, the focus should be on the children examining the properties of each 2-D shape, describing it according to these properties and contrasting it with, and comparing it to, the properties of other shapes, rather than on just naming the shape. For example, what makes a square a square? How is a square similar to, or different from, a rectangle? Could an argument be made to say a square is also a rectangle? Could an argument be made to say a rectangle is also a square?

Therefore, any new 2D shapes that the children encounter should be compared to the 2D shapes with which they are already familiar. And, as the 2D shapes chapter in Operation Maths always follows on from the topic of Lines and Angles, when exploring properties, reference should also be made to the number and type of angles within the shape, the number and types of sides (parallel, perpendicular etc) and whether the shape is regular or irregular.

Common misconceptions:

Categorising 2D shapes  separately: As mentioned previously, children often don’t recognise a square as a type of rectangle, a rectangle as a type of parallelogram, a rhombus as a type of parallelogram etc. This can often be the case if the children are focused primarily on naming the shape and then compartmentalising it in a category, as opposed to examining its properties and exploring how it may have proprieties in common with other shapes.

It can be useful here to display 2D shapes to the class using a subgroup structure (like this one here) so that the children can appreciate how, for example, a square is also a rectangle, is also a parallelogram, is also a quadrilateral, is also a polygon.

Constancy of shapes: Many children don’t recognise that a  shape remains constant, irrespective of its placement in space. In particular, a square sitting on its vertex is often incorrectly labelled as a diamond. The children should be encouraged to draw or trace around shapes on their MWBs and then rotate the shape in order to appreciate its constancy.

Regularity: Children may not recognise a five-sided figure with sides of same length as being a regular shape. It is as if the terminology “regular” implies to them that the shape should be common-place i.e. regularly occurring. For this same reason, a child will often say a rectangle and a circle are regular shapes, given their familiarity with these shapes from the junior classes, even though they are officially classified as irregular shapes. Challenging this misconception will require plenty of sorting activities where shapes are classified as regular or irregular (see Ready to Go activities below).

When creating a class display of shapes eg rectangles, triangles, etc., instead of using just one qualifying shape to illustrate the term, use many and use varied ones. Enlist the help of the class: “I want to make a display of triangles/parallelograms but I want the triangles/parallelograms to all be different. Can you draw and cut some out for me?” Such an activity would quickly reveal those who appreciate the required properties for each shape and those who don’t. Remember to also position the shapes in various ways so as to reinforce that the shape remains constant, irrespective of placement.

Identifying 3D objects as 2D shapes: This is a very problematic area. It often happens that when asked to find a circle in the environment, a child suggests a ball (a sphere) or a cube might be suggested as a square. When asking to identify 2D shapes in the classroom or at home, we must be careful how we respond to the answers so as not to reinforce these misconceptions. For example, if a child suggests that the door is a rectangle, when it is in fact a cuboid, emphasise that a part of the door is rectangular eg

• Which part of the door is like a rectangle?
• Are there any other parts of the door that are like a rectangle?
• Can you see any other rectangles on the outside of the door? How many?
• Are they all the same or different?

Asking the children to draw around solid 3D objects in order to produce flat 2D shapes can also be useful here.

Coordinates (6th class)

The concept of plotting and reading coordinates is introduced in 6th Class. There are plenty of examples of coordinates in the children’s environment, e.g. map reading, car parks and board games such as chess and Battleship. Allow the children to practise reading coordinates on maps and on board games, then progress to using two digit coordinates in maths. Make sure they first read the horizontal coordinate and then the vertical coordinate.

Operation Maths Digital Resources:

Don’t forget to access the linked digital activities on the digital version of the Pupil’s book, available on edcolearning.ie . These include:

Ready to go Activities: based on the Sorting eManipulative, these enable the various shapes to be sorted according to class-appropriate criteria, or enable tessellating patterns to be made. The Ready to go activities all have suggested questions inbuilt on the left-hand side of the screen that the teacher can just click to reveal and hide. Remember, when sorting, the focus should be on the properties of the shapes not their names; that said, you can also ask the children to identify the shapes, if known, as an extra dimension to the activity.

Create activities: (all classes) again using the Sorting eManipulative, these are less structured that their Ready to go counterparts. Instead, the teacher should click on the yellow “Create new example” button on the bottom of the screen, and then use the sorting eManipulative to explore the shapes as they see fit. The teacher can use a previous Ready to go activity to inspire the create activity or come up with a completely different activity of their own using the almost limitless possibilities of the sorting eManipulative.

Write-Hide-Show videos: These explore tessellations (3rd class) and different types of triangles (5th class). They encourage the children to look and respond to the questions by answering orally or on their MWBs.

Maths Around Us video (6th class): which examines different types of triangles from the environment.

Scratch-based programming lessons with instructions on how to draw 2D shapes  (3rd class) and hexagons (3rd, 4th, 5th classes), scalene triangle, pentagon and octagon (4th class), different types of triangles (5th class) and plot 2D shapes on a grid (6th class).

Further Reading and Resources

• Dear Family, your Operation Maths Guide to 2-D Shapes includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand unit. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.
• Shape and Space Manual from PDST
• 2D shapes Pinterest Board